Interference Alignment and Degrees of Freedom Region of Cellular Sigma Channel

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1 2011 IEEE Internatona Symposum on Informaton Theory Proceedngs Interference Agnment and Degrees of Freedom Regon of Ceuar Sgma Channe Huaru Yn 1 Le Ke 2 Zhengdao Wang 2 1 WINLAB Dept of EEIS Unv. of Sc. and Tech. of Chna Hefe Anhu P.R. Chna. ema: yhr@ustc.edu.cn 2 Dept. of ECE Iowa State Unversty Ames IA USA. ema: {keezhengdao@astate.edu Abstract We nvestgate the Degrees of Freedom DoF) Regon of a ceuar network where the ces can have overappng areas. Wthn an overappng area the mobe users can access mutpe base statons. We consder a case where there are two base statons both equpped wth mutpe antennas. The mobe statons are a equpped wth snge antenna and each mobe staton can beong to ether a snge ce or both ces. We competey characterze the DoF regon for the upnk channe assumng that goba channe state nformaton s avaabe at the transmtters. The achevabty scheme s based on nterference agnment at the base statons. I. INTRODUCTION Tradtona ceuar systems orthogonaze the channes such that sgnas sent from dfferent transmtters are supposed to be orthogona at east n the dea case n tme frequency or code dmensons. Such orthogonazaton yed technooges such as TDMA FDMA or CDMA. However orthogonazaton s not the most effcent way of utzng the avaabe sgna dmensons. Ths can be ceary seen even n a smpe twouser scaar Gaussan mutpe access channe: TDMA/FDMA s optma ony n one case where the bandwdth aocaton s proportona to the power avaabe to the users. For any other cases/rates orthogonazaton s strcty suboptma [2 Fg. 15.8]. Non-orthogona transmssons necessary create nterference at the recevers. How to desgn or contro such nterference s the key to hgher network effcency. Whe t s mportant to consder nterference the fuy couped nterference channe mode may be too pessmstc. The reason s that for ceuar networks the users we nsde a ce have hgh sgna to nterference rato SNR) and as a resut nterference can be negected compared to the usefu sgna. Whereas for users on the boundary of two or mutpe ces the nterference s comparabe wth the sgna. For nterference networks nstead of tryng to characterze the capacty regon competey whch s a dffcut probem the noton of degrees of freedom DoF) has been used to characterze how capacty scaes wth transmt power as the SNR goes to nfnty [4] [6]. The degrees of freedom s aso known as the mutpexng gan [10]. The nterference agnment for ceuar network has been consdered n [8] based on decomposabe channe where t s shown that the nterference free DoF can be acheved when the number of mobe statons ncreases. Practca usage of nterference agnment n ceuar network has been consdered n [7] [9]. In ths paper we consder a ceuar network that has overappng ces. Wthn an overappng area the mobe statons MS) can access mutpe base statons BS). Ths s a typca scenaro n ceuar communcatons. We consder a smpe case where there are ony two base statons both equpped wth mutpe antennas. We assume that the mobe statons are a equpped wth snge antenna. Our man contrbuton of the paper s the compete characterzaton of the upnk DoF regon for a ceuar system wth two base statons servng two overappng areas. The achevabty scheme uses nterference agnment at the two base statons. As a speca case of our resut we obtan the DoF regon of an X-network [1] wth snge antenna at the transmtters and mutpe antennas at two recevers. The rest of the paper s organzed as foows. In Secton II we present the system mode of the probem consdered. The statement of our man resut s presented n Secton III. The proof of the converse s presented n Secton IV and the achevabty s estabshed n Secton V. Fnay Secton VI concudes the paper. II. SYSTEM MODEL We consder a typca communcaton scenaro n wreess ceuar system where there s an overappng area between two ces; see Fg. 1. For smpcty we ony consder a system of two base statons. We assume the number of antennas at two base statons are N 1 and N 2 respectvey and a the mobe statons have snge antenna. Dependng on the ocatons of the mobe statons we dvde the mobe statons nto three groups: 1) Group A: mobe statons that communcate wth BS 1 ony; 2) Group B: mobe statons that communcate wth both BS 1 and BS 2; 3) Group C: mobe statons that communcate wth BS 2 ony. Denote the number of mobe statons n dfferent groups as L a L b and L c respectvey. We consder the upnk transmsson where mobe statons n Group A or Group C each have ony one ndependent message and mobe statons n Group B generate ndependent messages for both base statons. Therefore the tota number of messages s L = L a + 2L b + L c. We term such a channe as Σ channe due to the resembance see Fg. 2. Let t be tme ndex. The channe from the mobe staton j n Group A to BS 1 s denoted as h a) t) CN1 1 1 j L a. The channe from the mobe staton j n Group B to BS s denoted as h t) CN 1 {1 2 1 j L b. The channe from the mobe staton j n Group C to BS 2 s denoted as h c) 2j t) CN2 1 1 j L c. The channe coeffcents n dfferent tme nstants are a ndependenty and dentcay /11/$ IEEE 21

2 Ce 1 Ce 2 N 1 BS1 N 2 BS2 BS1 BS2 Type-A MS L a Type-B MS L b Type-C MS L c Fg. 1. Ceuar network wth overappng ces. Fg. 2. The Σ channe mode. generated from some contnuous dstrbuton whose mnmum and maxmum vaues are fnte. When {1 2 denote the ndex for BS we use c := 3 to refer to the ndex of the other BS. Denote the messages and the correspondng transmtted sgnas from the three types of mobe statons as W a) xa) t) 1 j L a W {1 2 1 j L b x j t) 1 j L b W c) 2j xc) 2j t) 1 j L c respectvey. The receved sgnas of BS 1 and 2 are as foows L a y 1 t) = h a) t)xa) t) + h t)x j t) + z 1 t) 1) L c y 2 t) = h c) 2j t)xc) 2j t) + h 2j t)x j t) + z 2 t) 2) where z t) C N 1 {1 2 are the addtve CN0 1) nose. The power of a transmtted sgnas s mted to P. The achevabe rates of the messages are denoted as R a) P) 1 j L a R P) {1 2 1 j L b and R c) 2j P) 1 j L c respectvey. The assocated DoF of the messages are d a) R a) = m P) P ogp) 1 j L a R = m P) P ogp) {1 2 1 j L b d c) 2j = m R c) 2j P) P ogp) 1 j L c. Let R L P) R + L denote the vector contanng a the rates at power P and d L R + L denote the vector contanng a the DoF. Let CP) R + L denote the capacty regon of the system whch contans a the rate tupes R L P) such that the probabty of error at a recevers can approach zero as the codng ength tends to nfnty. The DoF regon of the Σ channe s the coecton of a the DoF ponts D := { d L R + L : R LP) CP) such that d L = m P R L P) ogp). III. MAIN RESULT The man resut of our paper s the foowng theorem. Theorem 1: The DoF regon of Σ channe wth goba channe state nformaton at transmtters s specfed by the foowng nequates L a L c d a) 1 1 j L a; 3) + d 2j 1 1 j L b; 4) d c) 2j 1 1 j L c; 5) d a) + + 2j N 1 j J 2 J 2 {1 2 L b J 2 N 1 ; 6) d c) 2j + 2j + N 2 j J 1 J 1 {1 2 L b J 1 N 2. 7) IV. PROOF OF THE CONVERSE We prove the converse part of Theorem 1 n ths secton. The frst three nequates 3) 5) are due to the fact that a the mobe statons have snge antennas. To show the upper bound 6) we frst dvde the mobe statons n Group B nto two parts. We treat the mobe statons of Group B whose ndces n J 2 as a super user S b whose messages are W W 2j j J 2 and number of antennas s J 2. We treat the remanng mobe statons n Group B and the mobe statons n Group A as another super user S a. In addton we assume a the messages W 2j = 0 j / J 2. Therefore the super user S a has L a +L b J 2 antennas and ts messages are W a) 1 j L a and W 1 j L b j / J 2. We further assume that the mobe statons wthn S a and S b can fuy cooperate wth each other f they beong to the same super user set. Then S a S b BS 1 and BS 2 form a mutpe-nput and mutpe-output Z nterference channe. Snce cooperaton does no harm to the degrees of freedom the nequaty 6) hods based on [4 Coroary 1]. The nequaty 7) can be proved smary. V. ACHIEVABILITY OF THE DOF REGION We here gve the achevabty scheme of the DoF regon of Σ channe based on nterference agnment over the tme expanson channe. The proof s but upon the agnment scheme proposed n [3]. 22

3 A. Tme expanson modeng For any ratona DoF pont d L wthn D and satsfyng 3) 7) we can choose a postve nteger µ 0 such that µ 0 d L Z L + 8) where Z L + denotes the set of L-dmensona non-negatve ntegers. For any rratona DoF pont n the DoF regon we can aways approxmate t as a ratona pont wth arbtrary sma error. Denote µ n as the duraton n number of symbos of the tme expanson. Here and after we use the notaton to denote the tme expanded sgnas. Hence H = dagh 1)h 2)h µ n)) whch s a sze a) c) N µ n µ n bock dagona matrx. Matrces H and H 2j can be smary defned. For BS 1 we have the foowng two cases: 1) When L b > N 1 we w agn L b N 1 nterference messages at BS 1. For any DoF pont d L wthn D et J 2 denote a set contanng the ndces of mobe statons n Group B such that J 2 = N 1 and 2j 2j J 2 {1 2 L b J 2 N 1. j J 2 j J 2 Furthermore et { δ 2 = mn j j J 2 and 2j = mn k J 2k 2. 9) As we w see the nterference messages W 2j j J 2 w span the nterference space at BS 1 after gong through the tme-varyng channe. Among a these messages W s the message havng smaest DoF. For any other messages W J 2 ts DoF must be 2j j / ess than. We w agn these messages to message W 2j j J 2 at BS1. 2) When L b N 1 choose J 2 = {1...L b 6) becomes j J 1 L a d a) + j J 1 + d 2j ) N 1 10) whch suggests that a the messages are decodabe at BS 1. Therefore there s no need to do nterference agnment for BS 1. Smary. For BS 2 we have 1) when L b > N 2 et J1 denote the set contanng the ndces of mobe statons n Group B such that J 1 = N 2 and J 1 {1 2 L b J 1 N 2. In addton et { δ 1 = mn j j J 1 and = mn k J 1k 1. 11) 2) when L b N 2 there s no need to do agnment. Let Γ 1 = N 1 maxl b N 1 0) and Γ 2 = N 2 maxl b N 2 0). We sha see that they are the numbers of agnment constrants that mobe statons n Group B need to satsfy n order to agn nterference at BS 1 and BS 2 respectvey. We propose to use µ n = µ 0 n + 1) Γ1+Γ2 fod tme expanson where n s a postve nteger. Specfcay we want to acheve the foowng DoF over µ n sots d a) = µ 0 n Γ1+Γ2 d a) 1 j L a 12) = µ 0n Γ1+Γ2 1 j L b j / J 1 13) = µ 0n Γ1 n + 1) Γ2 1 j L b j J 1 14) 2j = µ 0n Γ1+Γ2 2j 1 j L b j / J 2 15) 2j = µ 0n + 1) Γ1 n Γ2 2j 1 j L b j J 2 16) d c) 2j = µ 0n Γ1+Γ2 d c) 2j 1 j L c. 17) Therefore when n the desred DoF pont d L can be acheved. The key s to desgn the beamformng coumn sets for mobes statons n Group B such that the DoF 12) 17) can be acheved over µ n sots. B. Beamformng and nterference agnment When L b N 1 no nterference agnment s needed at BS 1 n whch case we choose Ṽ 2j to be a sze µ n d 2j random fu rank matrx. Smary when L b N 2 we choose Ṽ to be a sze µ n d random fu rank matrx. In the foowng we assume that L b > maxn 1 N 2 ) and w desgn the beamformng matrces. We choose the beamformng coumn sets of mobe statons n Group B whose ndces beong to J 1 and J 2 to have the foowng forms Ṽ = [ P 11 ] j J 1 18) Ṽ 2j = [ P 21 where s a sze µ n 2j s a sze µ n d 2j 2j ] j J 2 19) d d ) random matrx and d ) random matrces. Obvousy Ṽ δ = P 1 { ) The two matrces P 11 and P 21 are structured and w be determned ater. In our desgn 18) and 19) a the messages W j J 1 sharng part of the same beamformng coumns whch s P 11. Smary a the messages W 2j j J 2 sharng part of the same beamformng coumns whch s P 21. For = 1 2 by the defnton of J the DoF of message W j / J s at most the same as that of W δ. It woud therefore be suffcent f we coud desgn a beamformng coumn set denoted as P 2 to be used by mobe staton j / J whch s abe to dever message wth DoF d δ. Denote the set of eements of J as {β 1 β 2...β N c. As the channe are random generated the foowng channes H 1) = [ H 21 H H 2) = [ H H 12 H 2N1 ] 21) H 1N2 ] 22) have fu rank wth probabty 1. As observed n [3] t s mpossbe to agn nterference message W j / J from 23

4 mobe staton j to ony one nterference message W k J k at BS c because the channe between any N c mobe statons to BS c are near ndependent wth probabty one. However we can choose H P22 [ H P21 21 H P21 22 H P21 2N1 ] j / J 2 H 2j P12 [ H P11 11 H P11 12 H P11 1N2 ] j / J 1 23) 24) so that the nterference space at BS c s not arger than that spanned by messages n J. We defne the µ n µ n matrces T ) = N accordng to the foowng T ) 1 T ) 2.. T ) N = H) ) 1 H j / J c. 25) It has been shown n [3] that these T ) matrces are dagona matrces. We aso defne the foowng bock dagona matrces P 1) = dag P 11 P {{ P 11 ) 26) N 2 matrces P 2) = dag P 21 P {{ P 21 ). 27) N 1 matrces It foows from 23) and 24) that T ) 1 T ) 2.. T ) N P c 2 P c) j / J c. 28) Therefore the agnment constrants for BS 1 and 2 are T ) P 22 P 21 j / J 2 1 N 1 29) P 12 P 11 j / J 1 1 N 2 30) T 2j) and the tota number of constrants s Γ 1 and Γ 2 respectvey. Denote B 1 = µ 0 n Γ1 and B 2 = µ 0 n Γ2. The matrces P 11 P 12 P 21 P 22 are desgned n 31) 34). It can be verfed that the number of coumns of P11 and P 21 are µ 0 n Γ1 n + 1) Γ2 and µ 0 n + 1) Γ1 n Γ2 respectvey. Therefore messages W and W can acheve desred DoF over µ n sots when n. It can be verfed that f message W j / J use P 2 as the beamformng matrx ts sgna w fa nto the nterference subspace spanned by messages W j J at BS c. That s a the agnment condtons n 29) and 30) are satsfed. Havng specfed the matrces P 11 P 12 P 21 P 22 we descrbe the beamformng matrces of a mobe statons n the foowng. 1) For mobe staton j n Group B f j J 1 t uses the beamformng matrx n 18) to transmt message W d to BS 1. Mobe staton j / J 1 randomy chooses coumns of P 12 as the beamformng matrx. Smary f j J 2 mobe staton j uses the beamformng matrx n 19) to transmt message W 2j to BS 2. mobe staton j / J 2 randomy chooses d 2j coumns of P 22 as the beamformng matrx. 2) Each mobe staton j n Group A randomy generates a) a µ n d matrx as beamformng matrx Ṽa). 3) Each mobe staton j n Group C randomy generates a c) µ n d 2j matrx as beamformng matrx Ṽc) 2j. 4) A the entres of the random beamformng coumns of mobe statons n Group A Group B and Group C are ndependenty and dentcay generated from some contnuous dstrbuton whose mnmum and maxmum vaues are fnte. C. Fu Rankness To guarantee that both base statons can decode the desred messages we frst need to make sure that a the beamformng matrces are fu rank whch s easy to verfy. In addton we need to guarantee the foowng condtons 1) [Ṽ Ṽ 2j ] has fu coumn rank for any mobe staton j n Group B. 2) The nterference space and sgna space are ndependent for both base statons. The frst condton s needed to guarantee that two messages of mobe statons n Group B can be dstngushed. It can be verfed that [ P 11 P 21 ] has µ 0 n Γ1 n + 1) Γ2 + µ 0 n + 1) Γ1 n Γ2 coumns whch may be arger than the number µ n of rows. However the beamformng matrx for any mobe staton w contan a subset of the coumns of [ P 11 P 21 ] pus possby some addtona random coumns. Snce 2j + 1 [Ṽ Ṽ 2j ] s aways a ta matrx more rows than coumns). To estabsh ts fu rankness note that each entry s a monoma the random varabes that defne the monomas are dfferent for a rows due to the tme varyng channe. In addton n one row the exponents of the monomas are dfferent. Therefore the condtons n Lemma 1 are satsfed and the sub-matrx has fu rank. We then need to vadate the second condton whch s needed to guarantee that the mobe statons can decode the messages that they are nterested n. We ony show ths for BS 1 as the same argument can be apped to BS 2 as we. Defne the foowng matrx 2) = [ dag N1 1 ) ] 35) where 0 s an a zero matrx. When L b > N 1 we woud ke to show that the matrx n 36) has fu coumn rank. Here A and B correspond to the sgna part whe C corresponds to the nterference space generated by the mobe statons whose ndces are n J 2. The number of coumns of A s µ 0 n Γ1+Γ2 L a. The number of coumns of B s da) µ 0 n Γ1+Γ2 L b j / J 1 + µ 0n Γ1 n + 1) Γ2 j J 1. The number of coumns of C s µ 0 n + 1) Γ1 n Γ2 j J 2 2j. 24

5 P 11 = P 12 = P 21 = P 22 = N 2j / J 1 1 N 2j / J 1 1 N 1j / J 2 1 N 1j / J 2 T 2j) T 2j) T ) T ) ) 2j) α 1 µn α 2j) {mn + m + 1 mn + m + 2 m + 1)n + m ) {mn + m + 1 mn + m + 2 m + 1)n + m 32) {mn + m + 1 mn + m + 2 m + 1)n + m ) {mn + m + 1 mn + m + 2 m + 1)n + m 34) ) 2j) α 1 µn α 2j) ) ) α 1 µn α ) ) ) α 1 µn α ) a) Λ 1 = H 11 Ṽa) a) 11 H 12 Ṽa) a) 12 H 1L a Ṽ a) 1L {{ a A H 11 Ṽ 11 H 12 Ṽ 12 H 1L b Ṽ 1L {{ b B H 1) P2) H 1) 2) {{ C 36) Therefore the number of coumns of Λ 1 s ess than µ 0 n + 1) Γ1+Γ2 N 1 due to 6). Hence Λ 1 s a ta matrx. We can see that the condtons of Lemma 1 st hod due to the foowng reasons: 1) A the eements of Λ 1 are monomas of dfferent random varabes. 2) The random varabes of dfferent rows are dfferent. 3) The random varabes of V a) do not appear n B and C. The random varabes n H do not appear n C. The random varabes n P 2) and 2) do not appear n A and B. Therefore the assocated exponents of monomas n one row dffers at east by one. Based on ths we concude that Λ 1 has fu coumn rank and the sgna space s ndependent from nterference space at BS 1. The proof of the achevabty s then compete. VI. CONCLUSION In the paper we proposed a Sgma Σ) channe mode for ceuar communcaton networks wth overappng ce areas. We aowed the base statons to have mutpe antennas and the mobe statons to have snge antennas. We derved the degrees of freedom regon for the upnk communcaton n the smpe ceuar network of two base statons under the assumpton that goba channe state nformaton s avaabe at the transmtters. The achevabty scheme s based on beamformng at the mobe statons and nterference agnment at the base statons. APPENDIX I Lemma 1: [1 Lemma 1] Consder an M M square matrx A such that a the eements n the th row and jth coumn of A s of the form where x [k] α [k] a = K k=1 x [k] ) α [k] 37) are random varabes and a exponents are ntegers Z. Suppose that 1) x [k] {x [k ] k) k ) has a contnuous cumuatve probabty dstrbuton. 2) j j { M wth j j ) α [1] α[2] α[k] α[2] α[k]. 38) α [1] ) In other words each random varabe has a contnuous cumuatve probabty dstrbuton condtoned on a the remanng varabes. Aso any two terms n the same row of the matrx A dffer n at east one exponent. Then the matrx A has a fu rank of M wth probabty 1. REFERENCES [1] V. Cadambe and S. Jafar Interference agnment and the degrees of freedom of wreess X networks IEEE Trans. Inform. Theory vo. 55 no. 9 pp Sept [2] T. M. Cover and J. A. Thomas Eements of Informaton Theory John Wey & Sons Inc. 2nd ed. edton [3] T. Gou and S. A. Jafar Degrees of freedom of the K user M N MIMO nterference channe 2008; [downoadabe from [4] S. Jafar and M. Fakhereddn Degrees of freedom for the MIMO nterference channe IEEE Trans. Inform. Theory vo. 53 no. 7 pp Juy [5] S. Jafar and S. Shama Degrees of freedom regon of the MIMO X channe IEEE Trans. Inform. Theory vo. 54 no. 1 pp Jan [6] M. A. Maddah-A S. A. Motahar and A. K. Khandan Communcaton over MIMO X channes: Interference agnment decomposton and performance anayss IEEE Trans. Inf. Theory no. 8 pp Aug [7] C. Suh M. Ho and D. Tse Downnk Interference Agnment n Proc. of IEEE GLOBECOM pp [8] C. Suh and D. Tse Interference Agnment for Ceuar Networks n Communcaton Contro and Computng th Annua Aerton Conference on pp [9] R. Tresch and M. Guaud Custered nterference agnment n arge ceuar networks n IEEE Internatona Symposum on Persona Indoor and Mobe Rado Communcatons pp [10] L. Zheng and D. N. C. Tse Dversty and mutpexng: a fundamenta tradeoff n mutpe-antenna channes IEEE Trans. Inf. Theory vo. 49 no. 5 pp May